We prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption. c pD 1 q has law Γ D 1 and ph D 1 D , φq " 0 a.s. for any φ with Supportpφq Ă DzD 1 .Remark 1.2. Note that in the domain Markov property, we have (by linearity) that if D 1 Ă D is simply connected, and φ 1 " φ 2 on D 1 , then ph D 1 D , φ 1 q " ph D 1 D , φ 2 q almost surely. When we discuss the domain Markov property later in the paper, we will often simply say that

We study how the Gaussian multiplicative chaos (GMC) measures µ γ corresponding to the 2D Gaussian free field change when γ approaches the critical parameter 2. In particular, we show that as γ → 2 − , (2 − γ) −1 µ γ converges in probability to 2µ ′ , where µ ′ is the critical GMC measure. 1 In the physics literature, "Liouville measure" refers to a volume form coming from a conformal field theory with a non-zero interaction term (see [RV16, Section 3.6]), inducing a certain weight on the law of the underlying GFF. Therefore, our measures correspond to a degenerate case, where the interaction parameter is equal to 0.2 Indeed, when one considers the approximation of the GFF on [0, 1] 2 by DGFF-s, then the conditions (4), (7) and the condition on exponential moments can be checked directly; the condition (5) follows from the estimates on the discrete Green's function due to Kenyon, as given for example in Theorem 2.5 of [CS11] 1 2

We show that, for general convolution approximations to a large class of log-correlated fields, including the 2d Gaussian free field, the critical chaos measures with derivative normalisation converge to a limiting measure µ ′ . This limiting measure does not depend on the choice of approximation. Moreover, it is equal to the measure obtained using the Seneta-Heyde renormalisation at criticality, or using a white-noise approximation to the field. AS14] one can hope to renormalise at criticality in one of two different ways. The first is called the Seneta-Heyde renormalisation, and involves premultiplying the sequence of measures (1.1) by the deterministic sequence log(1/ε). The other is a random renormalisation, which is defined by taking a derivative of the measure (1.1) in γ. It has been shown in [DRSV14a, DRSV14b] that for a special class of fields h having so-called ⋆-scale invariant kernels, and for a specific sequence of approximations to h, both procedures yield the same non-zero limiting measure (up to a constant). However, the result in these papers relies heavily on the cut-off approximation used for the kernel of h, and does not generalise to arbitrary convolution approximations. These are somewhat more natural, local approximations to the field, and the goal of the paper will be to extend the theory to this set-up.In this paper we will be particularly, but not exclusively, interested in the specific case where the underlying field h is a 2d Gaussian free field with zero-boundary conditions. In this case the measure µ γ (when it is defined and non-zero) is known as the Liouville measure with parameter γ. This has been an object of considerable recent interest due to its strong connection with 2d Liouville quantum gravity and the KPZ relations [DS11, RV11, Ber]. Recent works in the case γ < 2 include [DS11, RV11, Ber17], which among other things make an in-depth study of its moments, multifractal structure, and universality. Recently, in [APS], it has also been shown that these measures can be approximated using so-called local sets of the Gaussian free field. This is a particularly natural construction because it is both local and conformally invariant.The critical case γ = 2 has also been considered for the Gaussian free field: [DRSV14b, HRV, JS17, APS]. In [DRSV14b], the authors generalised their construction for ⋆-scale invariant kernels to show convergence in the Seneta-Heyde and derivative renormalisations for a specific "white noise" approximation to the field. These both yield the same (up to a constant) non-trivial limiting measure µ ′ , that we will call the critical Liouville measure. However, this proof again does not extend to convolution approximations.The purpose of this article is to complete the picture for convolution approximations to critical chaos. We will focus specifically on the case of the 2d GFF, and fields with ⋆-scale invariant kernels (in any number of space dimensions). This builds on recent work of Junnila and Saksman [JS17] (and also [HRV] in the case of the free field),...

We study branching diffusions in a bounded domain D of R d in which particles are killed upon hitting the boundary ∂D. It is known that any such process undergoes a phase transition when the branching rate β exceeds a critical value: a multiple of the first eigenvalue of the generator of the diffusion. We investigate the system at criticality and show that the associated genealogical tree, when the process is conditioned to survive for a long time, converges to Aldous' Continuum Random Tree under appropriate rescaling. The result holds under only a mild assumption on the domain, and is valid for all branching mechanisms with finite variance, and a general class of diffusions.We will prove that this sequence converges in distribution to a conditioned Brownian continuum random tree as t → ∞, with respect to the Gromov-Hausdorff topology. Indeed, if we let e be a Brownian excursion conditioned to reach height at least 1, and write (T e , d e ) for the real tree whose contour function is given by e, then we obtain the following result.where ϕ is the first eigenfunction of −L on D (see Section 2.1). Then, at the critical branching rate β = λ/(m − 1), and for any starting point x ∈ D,in distribution, with respect to the Gromov-Hausdorff topology.Remark 1.2. One sufficient condition to ensure that the hypotheses of Theorem 1.1 are satisfied is to assume that D is C 2,α for some α ∈ [0, 1] (see Lemma 2.3). However, this is also satisfied in many other cases, so we leave the assumptions of Theorem 1.1 as general as possible.On the way to proving Theorem 1.1 we also obtain new proofs of several other results concerning critical branching diffusions, some of which are already known in various forms. The reason for detailing these proofs here is threefold: firstly, it allows us to pin down the regularity required on the domain D; secondly, it provides a new and somewhat more probabilistic approach to the theory, that we believe is interesting in its own right; and finally, the proofs serve to introduce many concepts and ideas that are crucial for the proof of Theorem 1.1.Let us first look at the phase transition. This result was originally proved by Sevast'yanov [Sev58] and Watanabe [Wat65], in the case when L is a constant multiple of ∆. However, it has also been reworked and generalised since then. In [AH83, Chapter 6], a more general version of the result is given for branching Markov processes whose moment semigroup satisfies a certain criterion. One of the main examples discussed is when the process is a branching diffusion on a manifold with killing at the boundary. This is slightly more general than the set up of the present paper, in that the diffusion is on a manifold and the branching mechanism is allowed to be spatially dependent, however, a (fairly abstract) condition on the moment semigroup is required. In [Her78b] the criterion is shown to be satisfied, for example, if the manifold has C 3 boundary and the generator of the diffusion is uniformly elliptic with C 2 coefficients. Here we will prove the result un...